While weak solutions are in general global, or in the case of fluid-structure interaction exist up to a collision or self-collision of elastic structures, they are very often not unique and have a low regularity that comes from the energy of the problem. On the other hand, strong (or more regular) solutions are usually unique and posses higher regularity, but are often local in time. Thus, a natural question arises - when do weak and strong solutions coincide if eminating from the same initial data, or in other words, when do weak solutions enjoy so called weak-strong uniqueness property? Note that this only gives a partial answer to the uniqueness problem, as it usually doesn't hold globally in time and requires a more regular initial data in order to ensure the existence of a strong solution in the first place. Some systems, like the incompressible Navier-Stokes equations, actually enjoy a better type of uniqueness properties. In particular, finite energy weak solutions in 2D are unique, while in 3D, a different type of a weaker uniquneness, so called Prodi-Serrin, is proven to hold. This means that a weak solution that has the integrability \(\mathbf {u}\in L_t^a L_x^b\) , for \(a\in (2,\infty ), b\in (3,\infty )\)  such that \(\frac{2}{a}+\frac{3}{b}=1\) , is unique in the class of weak solutions. On the other hand, the compressible Navier-Stokes system is not as fortunate, as only the results concerning the weak-strong uniqueness are available so far (see [1]}, [2]}, [3]}), at least without any further simplifications.
